Simplify ( square root of -75)/( square root of -72)

$\frac{\sqrt{- 75}}{\sqrt{- 72}}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 75$ as $- 1 (75)$ .

$\frac{\sqrt{- 1 (75)}}{\sqrt{- 72}}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$\frac{\sqrt{- 1} \cdot \sqrt{75}}{\sqrt{- 72}}$

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{i \cdot \sqrt{75}}{\sqrt{- 72}}$

Rewrite $- 72$ as $- 1 (72)$ .

$\frac{i \cdot \sqrt{75}}{\sqrt{- 1 (72)}}$

Rewrite $\sqrt{- 1 (72)}$ as $\sqrt{- 1} \cdot \sqrt{72}$ .

$\frac{i \cdot \sqrt{75}}{\sqrt{- 1} \cdot \sqrt{72}}$

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{i \cdot \sqrt{75}}{i \cdot \sqrt{72}}$

Step 2

Cancel the common factor of $i$ .

Cancel the common factor.

$\frac{i \cdot \sqrt{75}}{i \cdot \sqrt{72}}$

Rewrite the expression.

$\frac{\sqrt{75}}{\sqrt{72}}$

Step 3

Combine $\sqrt{75}$ and $\sqrt{72}$ into a single radical.

$\sqrt{\frac{75}{72}}$

Step 4

Cancel the common factor of $75$ and $72$ .

Factor $3$ out of $75$ .

$\sqrt{\frac{3 (25)}{72}}$

Cancel the common factors.

Factor $3$ out of $72$ .

$\sqrt{\frac{3 \cdot 25}{3 \cdot 24}}$

Cancel the common factor.

$\sqrt{\frac{3 \cdot 25}{3 \cdot 24}}$

Rewrite the expression.

$\sqrt{\frac{25}{24}}$

Step 5

Rewrite $\sqrt{\frac{25}{24}}$ as $\frac{\sqrt{25}}{\sqrt{24}}$ .

$\frac{\sqrt{25}}{\sqrt{24}}$

Step 6

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{24}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{5}{\sqrt{24}}$

Step 7

Simplify the denominator.

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$\frac{5}{\sqrt{4 (6)}}$

Rewrite $4$ as $2^{2}$ .

$\frac{5}{\sqrt{2^{2} \cdot 6}}$

Pull terms out from under the radical.

$\frac{5}{2 \sqrt{6}}$

Step 8

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{5}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 9

Combine and simplify the denominator.

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{5 \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Move $\sqrt{6}$ .

$\frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{5 \sqrt{6}}{2 ({\sqrt{6}}^{1} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{5 \sqrt{6}}{2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{5 \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{5 \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5 \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{5 \sqrt{6}}{2 \cdot 6^{1}}$

Evaluate the exponent.

$\frac{5 \sqrt{6}}{2 \cdot 6}$

Step 10

Multiply $2$ by $6$ .

$\frac{5 \sqrt{6}}{12}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{6}}{12}$

Decimal Form:

$1.02062072 \dots$

Simplify ( square root of -75)/( square root of -72)

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